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Eigen::ColPivHouseholderQR< _MatrixType > Class Template Reference
Dense linear problems and decompositions » Reference » QR_Module

Householder rank-revealing QR decomposition of a matrix with column-pivoting. More...

#include <ForwardDeclarations.h>

Public Types

enum  { RowsAtCompileTime = MatrixType::RowsAtCompileTime, ColsAtCompileTime = MatrixType::ColsAtCompileTime, MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime, MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime }
 
typedef internal::plain_diag_type< MatrixType >::type HCoeffsType
 
typedef HouseholderSequence< MatrixType, typename internal::remove_all< typename HCoeffsType::ConjugateReturnType >::type > HouseholderSequenceType
 
typedef internal::plain_row_type< MatrixType, Index >::type IntRowVectorType
 
typedef _MatrixType MatrixType
 
typedef PermutationMatrix< ColsAtCompileTime, MaxColsAtCompileTime > PermutationType
 
typedef MatrixType::PlainObject PlainObject
 
typedef internal::plain_row_type< MatrixType, RealScalar >::type RealRowVectorType
 
typedef MatrixType::RealScalar RealScalar
 
typedef internal::plain_row_type< MatrixType >::type RowVectorType
 
typedef MatrixType::Scalar Scalar
 
typedef MatrixType::StorageIndex StorageIndex
 

Public Member Functions

template<typename RhsType , typename DstType >
EIGEN_DEVICE_FUNC void _solve_impl (const RhsType &rhs, DstType &dst) const
 
template<typename RhsType , typename DstType >
void _solve_impl (const RhsType &rhs, DstType &dst) const
 
MatrixType::RealScalar absDeterminant () const
 
 ColPivHouseholderQR ()
 Default Constructor. More...
 
 ColPivHouseholderQR (Index rows, Index cols)
 Default Constructor with memory preallocation. More...
 
template<typename InputType >
 ColPivHouseholderQR (const EigenBase< InputType > &matrix)
 Constructs a QR factorization from a given matrix. More...
 
template<typename InputType >
 ColPivHouseholderQR (EigenBase< InputType > &matrix)
 Constructs a QR factorization from a given matrix. More...
 
Index cols () const
 
const PermutationType & colsPermutation () const
 
template<typename InputType >
ColPivHouseholderQR & compute (const EigenBase< InputType > &matrix)
 
template<typename InputType >
ColPivHouseholderQR< MatrixType > & compute (const EigenBase< InputType > &matrix)
 
Index dimensionOfKernel () const
 
const HCoeffsType & hCoeffs () const
 
HouseholderSequenceType householderQ () const
 
ComputationInfo info () const
 Reports whether the QR factorization was succesful. More...
 
const Inverse< ColPivHouseholderQR > inverse () const
 
bool isInjective () const
 
bool isInvertible () const
 
bool isSurjective () const
 
MatrixType::RealScalar logAbsDeterminant () const
 
HouseholderSequenceType matrixQ () const
 
const MatrixType & matrixQR () const
 
const MatrixType & matrixR () const
 
RealScalar maxPivot () const
 
Index nonzeroPivots () const
 
Index rank () const
 
Index rows () const
 
ColPivHouseholderQR & setThreshold (const RealScalar &threshold)
 
ColPivHouseholderQR & setThreshold (Default_t)
 
template<typename Rhs >
const Solve< ColPivHouseholderQR, Rhs > solve (const MatrixBase< Rhs > &b) const
 
RealScalar threshold () const
 

Protected Member Functions

void computeInPlace ()
 

Static Protected Member Functions

static void check_template_parameters ()
 

Protected Attributes

RealRowVectorType m_colNormsDirect
 
RealRowVectorType m_colNormsUpdated
 
PermutationType m_colsPermutation
 
IntRowVectorType m_colsTranspositions
 
Index m_det_pq
 
HCoeffsType m_hCoeffs
 
bool m_isInitialized
 
RealScalar m_maxpivot
 
Index m_nonzero_pivots
 
RealScalar m_prescribedThreshold
 
MatrixType m_qr
 
RowVectorType m_temp
 
bool m_usePrescribedThreshold
 

Private Types

typedef PermutationType::StorageIndex PermIndexType
 

Friends

class CompleteOrthogonalDecomposition< MatrixType >
 

Detailed Description

template<typename _MatrixType>
class Eigen::ColPivHouseholderQR< _MatrixType >

Householder rank-revealing QR decomposition of a matrix with column-pivoting.

Template Parameters
_MatrixTypethe type of the matrix of which we are computing the QR decomposition

This class performs a rank-revealing QR decomposition of a matrix A into matrices P, Q and R such that

\[ \mathbf{A} \, \mathbf{P} = \mathbf{Q} \, \mathbf{R} \]

by using Householder transformations. Here, P is a permutation matrix, Q a unitary matrix and R an upper triangular matrix.

This decomposition performs column pivoting in order to be rank-revealing and improve numerical stability. It is slower than HouseholderQR, and faster than FullPivHouseholderQR.

This class supports the inplace decomposition mechanism.

See also
MatrixBase::colPivHouseholderQr()

Definition at line 255 of file ForwardDeclarations.h.

Member Typedef Documentation

template<typename _MatrixType>
typedef internal::plain_diag_type<MatrixType>::type Eigen::ColPivHouseholderQR< _MatrixType >::HCoeffsType

Definition at line 63 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef HouseholderSequence<MatrixType,typename internal::remove_all<typename HCoeffsType::ConjugateReturnType>::type> Eigen::ColPivHouseholderQR< _MatrixType >::HouseholderSequenceType

Definition at line 68 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType, Index>::type Eigen::ColPivHouseholderQR< _MatrixType >::IntRowVectorType

Definition at line 65 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef _MatrixType Eigen::ColPivHouseholderQR< _MatrixType >::MatrixType

Definition at line 52 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef PermutationType::StorageIndex Eigen::ColPivHouseholderQR< _MatrixType >::PermIndexType
private

Definition at line 73 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> Eigen::ColPivHouseholderQR< _MatrixType >::PermutationType

Definition at line 64 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef MatrixType::PlainObject Eigen::ColPivHouseholderQR< _MatrixType >::PlainObject

Definition at line 69 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType, RealScalar>::type Eigen::ColPivHouseholderQR< _MatrixType >::RealRowVectorType

Definition at line 67 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef MatrixType::RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::RealScalar

Definition at line 60 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef internal::plain_row_type<MatrixType>::type Eigen::ColPivHouseholderQR< _MatrixType >::RowVectorType

Definition at line 66 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef MatrixType::Scalar Eigen::ColPivHouseholderQR< _MatrixType >::Scalar

Definition at line 59 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
typedef MatrixType::StorageIndex Eigen::ColPivHouseholderQR< _MatrixType >::StorageIndex

Definition at line 62 of file ColPivHouseholderQR.h.

Member Enumeration Documentation

template<typename _MatrixType>
anonymous enum
Enumerator
RowsAtCompileTime 
ColsAtCompileTime 
MaxRowsAtCompileTime 
MaxColsAtCompileTime 

Definition at line 53 of file ColPivHouseholderQR.h.

Constructor & Destructor Documentation

template<typename _MatrixType>
Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( )
inline

Default Constructor.

The default constructor is useful in cases in which the user intends to perform decompositions via ColPivHouseholderQR::compute(const MatrixType&).

Definition at line 83 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( Index  rows,
Index  cols 
)
inline

Default Constructor with memory preallocation.

Like the default constructor but with preallocation of the internal data according to the specified problem size.

See also
ColPivHouseholderQR()

Definition at line 100 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
template<typename InputType >
Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( const EigenBase< InputType > &  matrix)
inlineexplicit

Constructs a QR factorization from a given matrix.

This constructor computes the QR factorization of the matrix matrix by calling the method compute(). It is a short cut for:

ColPivHouseholderQR<MatrixType> qr(matrix.rows(), matrix.cols());
qr.compute(matrix);
See also
compute()

Definition at line 124 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
template<typename InputType >
Eigen::ColPivHouseholderQR< _MatrixType >::ColPivHouseholderQR ( EigenBase< InputType > &  matrix)
inlineexplicit

Constructs a QR factorization from a given matrix.

This overloaded constructor is provided for inplace decomposition when MatrixType is a Eigen::Ref.

See also
ColPivHouseholderQR(const EigenBase&)

Definition at line 145 of file ColPivHouseholderQR.h.

Member Function Documentation

template<typename _MatrixType>
template<typename RhsType , typename DstType >
EIGEN_DEVICE_FUNC void Eigen::ColPivHouseholderQR< _MatrixType >::_solve_impl ( const RhsType &  rhs,
DstType &  dst 
) const
template<typename _MatrixType>
template<typename RhsType , typename DstType >
void Eigen::ColPivHouseholderQR< _MatrixType >::_solve_impl ( const RhsType &  rhs,
DstType &  dst 
) const

Definition at line 585 of file ColPivHouseholderQR.h.

template<typename MatrixType >
MatrixType::RealScalar Eigen::ColPivHouseholderQR< MatrixType >::absDeterminant ( ) const
Returns
the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.
Note
This is only for square matrices.
Warning
a determinant can be very big or small, so for matrices of large enough dimension, there is a risk of overflow/underflow. One way to work around that is to use logAbsDeterminant() instead.
See also
logAbsDeterminant(), MatrixBase::determinant()

Definition at line 448 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
static void Eigen::ColPivHouseholderQR< _MatrixType >::check_template_parameters ( )
inlinestaticprotected

Definition at line 427 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::cols ( void  ) const
inline

Definition at line 328 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
const PermutationType& Eigen::ColPivHouseholderQR< _MatrixType >::colsPermutation ( ) const
inline
Returns
a const reference to the column permutation matrix

Definition at line 214 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
template<typename InputType >
ColPivHouseholderQR& Eigen::ColPivHouseholderQR< _MatrixType >::compute ( const EigenBase< InputType > &  matrix)
template<typename _MatrixType>
template<typename InputType >
ColPivHouseholderQR<MatrixType>& Eigen::ColPivHouseholderQR< _MatrixType >::compute ( const EigenBase< InputType > &  matrix)

Performs the QR factorization of the given matrix matrix. The result of the factorization is stored into *this, and a reference to *this is returned.

See also
class ColPivHouseholderQR, ColPivHouseholderQR(const MatrixType&)

Definition at line 472 of file ColPivHouseholderQR.h.

template<typename MatrixType >
void Eigen::ColPivHouseholderQR< MatrixType >::computeInPlace ( )
protected

Definition at line 480 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::dimensionOfKernel ( ) const
inline
Returns
the dimension of the kernel of the matrix of which *this is the QR decomposition.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 272 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
const HCoeffsType& Eigen::ColPivHouseholderQR< _MatrixType >::hCoeffs ( ) const
inline
Returns
a const reference to the vector of Householder coefficients used to represent the factor Q.

For advanced uses only.

Definition at line 334 of file ColPivHouseholderQR.h.

template<typename MatrixType >
ColPivHouseholderQR< MatrixType >::HouseholderSequenceType Eigen::ColPivHouseholderQR< MatrixType >::householderQ ( ) const
Returns
the matrix Q as a sequence of householder transformations. You can extract the meaningful part only by using:
qr.householderQ().setLength(qr.nonzeroPivots())

Definition at line 634 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
ComputationInfo Eigen::ColPivHouseholderQR< _MatrixType >::info ( ) const
inline

Reports whether the QR factorization was succesful.

Note
This function always returns Success. It is provided for compatibility with other factorization routines.
Returns
Success

Definition at line 411 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
const Inverse<ColPivHouseholderQR> Eigen::ColPivHouseholderQR< _MatrixType >::inverse ( ) const
inline
Returns
the inverse of the matrix of which *this is the QR decomposition.
Note
If this matrix is not invertible, the returned matrix has undefined coefficients. Use isInvertible() to first determine whether this matrix is invertible.

Definition at line 321 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
bool Eigen::ColPivHouseholderQR< _MatrixType >::isInjective ( ) const
inline
Returns
true if the matrix of which *this is the QR decomposition represents an injective linear map, i.e. has trivial kernel; false otherwise.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 285 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
bool Eigen::ColPivHouseholderQR< _MatrixType >::isInvertible ( ) const
inline
Returns
true if the matrix of which *this is the QR decomposition is invertible.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 310 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
bool Eigen::ColPivHouseholderQR< _MatrixType >::isSurjective ( ) const
inline
Returns
true if the matrix of which *this is the QR decomposition represents a surjective linear map; false otherwise.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 298 of file ColPivHouseholderQR.h.

template<typename MatrixType >
MatrixType::RealScalar Eigen::ColPivHouseholderQR< MatrixType >::logAbsDeterminant ( ) const
Returns
the natural log of the absolute value of the determinant of the matrix of which *this is the QR decomposition. It has only linear complexity (that is, O(n) where n is the dimension of the square matrix) as the QR decomposition has already been computed.
Note
This is only for square matrices.
This method is useful to work around the risk of overflow/underflow that's inherent to determinant computation.
See also
absDeterminant(), MatrixBase::determinant()

Definition at line 457 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
HouseholderSequenceType Eigen::ColPivHouseholderQR< _MatrixType >::matrixQ ( ) const
inline

Definition at line 182 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
const MatrixType& Eigen::ColPivHouseholderQR< _MatrixType >::matrixQR ( ) const
inline
Returns
a reference to the matrix where the Householder QR decomposition is stored

Definition at line 189 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
const MatrixType& Eigen::ColPivHouseholderQR< _MatrixType >::matrixR ( ) const
inline
Returns
a reference to the matrix where the result Householder QR is stored
Warning
The strict lower part of this matrix contains internal values. Only the upper triangular part should be referenced. To get it, use
matrixR().template triangularView<Upper>()
For rank-deficient matrices, use
matrixR().topLeftCorner(rank(), rank()).template triangularView<Upper>()

Definition at line 204 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::maxPivot ( ) const
inline
Returns
the absolute value of the biggest pivot, i.e. the biggest diagonal coefficient of R.

Definition at line 403 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::nonzeroPivots ( ) const
inline
Returns
the number of nonzero pivots in the QR decomposition. Here nonzero is meant in the exact sense, not in a fuzzy sense. So that notion isn't really intrinsically interesting, but it is still useful when implementing algorithms.
See also
rank()

Definition at line 394 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::rank ( ) const
inline
Returns
the rank of the matrix of which *this is the QR decomposition.
Note
This method has to determine which pivots should be considered nonzero. For that, it uses the threshold value that you can control by calling setThreshold(const RealScalar&).

Definition at line 255 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
Index Eigen::ColPivHouseholderQR< _MatrixType >::rows ( void  ) const
inline

Definition at line 327 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
ColPivHouseholderQR& Eigen::ColPivHouseholderQR< _MatrixType >::setThreshold ( const RealScalar &  threshold)
inline

Allows to prescribe a threshold to be used by certain methods, such as rank(), who need to determine when pivots are to be considered nonzero. This is not used for the QR decomposition itself.

When it needs to get the threshold value, Eigen calls threshold(). By default, this uses a formula to automatically determine a reasonable threshold. Once you have called the present method setThreshold(const RealScalar&), your value is used instead.

Parameters
thresholdThe new value to use as the threshold.

A pivot will be considered nonzero if its absolute value is strictly greater than $ \vert pivot \vert \leqslant threshold \times \vert maxpivot \vert $ where maxpivot is the biggest pivot.

If you want to come back to the default behavior, call setThreshold(Default_t)

Definition at line 353 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
ColPivHouseholderQR& Eigen::ColPivHouseholderQR< _MatrixType >::setThreshold ( Default_t  )
inline

Allows to come back to the default behavior, letting Eigen use its default formula for determining the threshold.

You should pass the special object Eigen::Default as parameter here.

qr.setThreshold(Eigen::Default);

See the documentation of setThreshold(const RealScalar&).

Definition at line 368 of file ColPivHouseholderQR.h.

template<typename _MatrixType>
template<typename Rhs >
const Solve<ColPivHouseholderQR, Rhs> Eigen::ColPivHouseholderQR< _MatrixType >::solve ( const MatrixBase< Rhs > &  b) const
inline

This method finds a solution x to the equation Ax=b, where A is the matrix of which *this is the QR decomposition, if any exists.

Parameters
bthe right-hand-side of the equation to solve.
Returns
a solution.

Example:

Matrix3f m = Matrix3f::Random();
Matrix3f y = Matrix3f::Random();
cout << "Here is the matrix m:" << endl << m << endl;
cout << "Here is the matrix y:" << endl << y << endl;
Matrix3f x;
x = m.colPivHouseholderQr().solve(y);
assert(y.isApprox(m*x));
cout << "Here is a solution x to the equation mx=y:" << endl << x << endl;

Output: 

 

Definition at line 175 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::threshold 
          (
          )
           const
        

      

  		
  
inline  
  






Returns the threshold that will be used by certain methods such as rank().


See the documentation of setThreshold(const RealScalar&). 



Definition at line 378 of file ColPivHouseholderQR.h.







Friends And Related Function Documentation








template<typename _MatrixType> 



  

  
      

        

          friend class CompleteOrthogonalDecomposition< MatrixType >
        

      

  		
  
friend  
  







Definition at line 425 of file ColPivHouseholderQR.h.







Member Data Documentation








template<typename _MatrixType> 



  

  
      

        

          RealRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colNormsDirect
        

      

  		
  
protected  
  







Definition at line 440 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          RealRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colNormsUpdated
        

      

  		
  
protected  
  







Definition at line 439 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          PermutationType Eigen::ColPivHouseholderQR< _MatrixType >::m_colsPermutation
        

      

  		
  
protected  
  







Definition at line 436 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          IntRowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_colsTranspositions
        

      

  		
  
protected  
  







Definition at line 437 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::ColPivHouseholderQR< _MatrixType >::m_det_pq
        

      

  		
  
protected  
  







Definition at line 444 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          HCoeffsType Eigen::ColPivHouseholderQR< _MatrixType >::m_hCoeffs
        

      

  		
  
protected  
  







Definition at line 435 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::ColPivHouseholderQR< _MatrixType >::m_isInitialized
        

      

  		
  
protected  
  







Definition at line 441 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::m_maxpivot
        

      

  		
  
protected  
  







Definition at line 442 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          Index Eigen::ColPivHouseholderQR< _MatrixType >::m_nonzero_pivots
        

      

  		
  
protected  
  







Definition at line 443 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          RealScalar Eigen::ColPivHouseholderQR< _MatrixType >::m_prescribedThreshold
        

      

  		
  
protected  
  







Definition at line 442 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          MatrixType Eigen::ColPivHouseholderQR< _MatrixType >::m_qr
        

      

  		
  
protected  
  







Definition at line 434 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          RowVectorType Eigen::ColPivHouseholderQR< _MatrixType >::m_temp
        

      

  		
  
protected  
  







Definition at line 438 of file ColPivHouseholderQR.h.













template<typename _MatrixType> 



  

  
      

        

          bool Eigen::ColPivHouseholderQR< _MatrixType >::m_usePrescribedThreshold
        

      

  		
  
protected  
  







Definition at line 441 of file ColPivHouseholderQR.h.







The documentation for this class was generated from the following files:







gtsam

Author(s): 

autogenerated on Sat May 8 2021 02:52:23